The paper presents a novel Reynolds-averaged turbulence model for flows of power-law fluid. The model uses the elliptic relaxation approach to capture the near-wall turbulence anisotropy. The turbulence model for Newtonians fluids is modified by introducing closed approximations of correlations between velocity and viscosity fluctuations. The approximation for non-Newtonian extra stress is derived with the assumption of smallness of molecular viscosity fluctuations. A closed model for the averaged molecular viscosity is derived which takes into account its nonlinear dependence on the shear rate. Validation of the model against the direct numerical simulation (DNS) data for power-law fluids flows in the pipe demonstrates that new model is able to predict the main features of the non-Newtonian turbulence. Mean velocity, turbulent energy and averaged molecular viscosity distributions agree well with DNS data. (c) 2015 Published by Elsevier B.V.